I think the machine should still start up without the battery. You'd just lose your clock settings when you shut down. According to MacTracker, your machine uses a 3.6v litium battery, but I don't know what size. Try to get the battery, perhaps it is indeed the root of the no-video problem.

That would indeed cause you to startup without video. I ran into the same problem with a 6100 about a year ago. After much gnashing of teeth and pulling of hair I replaced an old battery and video was restored. You can get a new battery at radio shack, or OWC has them online.

I had a 6100/66 with a dead battery for ages, the only thing it hurt was the clock settings. The thing seemed to always start up with no video. What I would do, even though this isn't prob the best idea, is turn on the switch til u hear the chime, then turn it off, then back on again and then the chime and video comes up. Worked for me for a while.

edit:
I did replace the battery eventually, don't recall if it helped the video at startup or not.. but I did get it from OWC for a few bux.

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